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In FOL, a formula is interpreted in meta-language as $m,s\models\phi$, where m is a model/structure and s is an assignment function. If we continue to interpret this meta-language sentence, how can we do? How to determine a domain? I think this domain should at least include some models, some assignment functions, and some FOL formulas. This seems quite different from the domain we choose to interpret an FOL formula, which contains things of the same kind. The objects in the domain for the meta-language belong to various kinds(models, assignment functions, FOL formulas, etc.) Am I on the wrong way?

William
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  • Not very clear... we read an expression of the metalanguage, like e.g. $M, s \vDash \phi$ as a "usual" math formula. – Mauro ALLEGRANZA Mar 19 '24 at 10:02
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    If we want to formalize metatheory, we can use Weak Second-order Arithmetic or Set Theory (with very limited "infinity claims"). Some post with details is for sure available in MSE. – Mauro ALLEGRANZA Mar 19 '24 at 10:06
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    See e.g. the post First-order semantics with minimal metatheory as well as this post – Mauro ALLEGRANZA Mar 19 '24 at 10:25
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    And also Difference between metatheory and theory as well as this paper – Mauro ALLEGRANZA Mar 19 '24 at 10:32
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    "The objects in the domain for the meta-language belong to various kinds: models, assignment functions, FOL formulas, etc." If we work inj set theory, models are mathematical structures, i.e. sets with operations, i.e. n-uple (i.e. sets) with a set and functions (i.e. sets) on it; an assignment function is a function (i.e. a set) and formulas are expressions, i.e. finite strings, i.e. n-uples. – Mauro ALLEGRANZA Mar 19 '24 at 10:39
  • @MauroALLEGRANZA so, various kinds of objects are allowed in the domain to interpret a meta-language sentence? – William Mar 19 '24 at 11:49
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    If $\phi$ is a first-order formula, $M$ is a structure, and $s$ is an assignment function, we have a precise definition of the satisfaction relation $M,s\models \phi$. What does it mean to "continue to interpret this meta-language sentence"? If you're wondering about semantics for the meta-language, what they will look like of course depends on what kind of meta-theory you're working with. The standard is a set theory like ZFC, in which concepts like "structure", "formula", and "assignment function" can all be encoded as certain sets. Is this anywhere close to the question you intended to ask? – Alex Kruckman Mar 19 '24 at 18:02
  • @AlexKruckman Yes, that is close to what I intended to ask. You know, when we interpret a FOL formula, we select one domain. However, to interpret a meta-language sentence, it seems natural to use multiple domains, i.e., a domain contains all models, a domain contains all FOL formulas. etc., because they are so different. If so, the framework of interpreting the meta-language would be different than the one of interpreting FOL. – William Mar 20 '24 at 02:37
  • "However, to interpret a meta-language sentence, it seems natural to use multiple domains" - I don't really understand what you mean. To me, it does not seem natural. After all, the meta-language may just as well be an ordinary first-order theory like ZFC set theory... – Alex Kruckman Mar 20 '24 at 02:53
  • @AlexKruckman The meta-language is much more complex than the set theory, for example, the symbol ⊨ is not in the set theory. – William Mar 20 '24 at 08:31
  • No, but $\models$ is a defined concept within set theory. ZFC is a foundational theory, i.e., it can serve as a foundation for mathematics, meaning that every other mathematical concept can be defined/implemented inside set theory. In particular, set theory can serve as the meta-theory when we're doing mathematical logic. At this point you should go read some of the many previous discussions of the meta-theory on this site, rather than continuing this comment thread. – Alex Kruckman Mar 20 '24 at 11:55
  • @AlexKruckman Maybe we are talking about two different things. In what you are talking, M,s⊨ϕ and the set theory are in the same meta-language, where ⊨ may be defined by other symbols of the set theory, while I want to talk about the interpretation of M,s⊨ϕ in its meta-theory(maybe we should call it meta-meta-theory). – William Mar 20 '24 at 14:33
  • @William I'm sorry, but I don't understand your last comment at all. I'm going to reiterate my previous comment that you should read up on what "meta-theory" actually means (on this site, or in books on foundations / philosophy of mathematics). I'm not going to continue this comment thread. – Alex Kruckman Mar 20 '24 at 14:46
  • Obviously the symbol $\vDash$ is not in the set theory but we can easily define a ternary relation $\text{Sat} (M,s,f)$ that holds between a model $M$ (a mathematical structure, i.e. aset with some operations) a function $s$ (that is a set) and a formula $f$ (a finite expression i.e. string of symbols). – Mauro ALLEGRANZA Mar 26 '24 at 15:08

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